Processor for entangled complex signals

ABSTRACT

A system for and method of processing complex signals encoded into quantum states is presented. According to an embodiment of the invention, polarized components of a pump laser beam are separated and respectively modulated with first and second signals. The modulated polarized components are directed to adjacent non-linear crystals with optical axes aligned at right angles to each-other. Information regarding at least one of the first and second signals is then derived from measurements of coincidence events.

This application claims priority to U.S. Provisional Application No.60/701,969 to Freeling entitled “PROCESSOR FOR ENTANGLED COMPLEXSIGNALS,” the contents of which are incorporated by reference herein intheir entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a system for and method of encoding classicalcomplex signals into quantum entangled states. In particular, theinvention relates to a general system for and method of quantuminformation processing.

2. Discussion of Background Information

Photons are quanta of electromagnetic energy. Multiple photons may beentangled or not entangled. Photons that are not entangled together(i.e., random photons) exist as independent entities. In contrast,entangled photons have a connection between their respective properties.

Two photons entangled together are referred to as an entangled-photonpair (also, “biphotons”). Traditionally, photons comprising anentangled-photon pair are called “signal” and “idler” photons. Measuringproperties of one photon of an entangled-photon pair determines resultsof measurements of corresponding properties of the other photon, even ifthe two entangled photons are separated by a distance. As understood bythose of ordinary skill in the art and by way of non-limiting example,the quantum mechanical state of an entangled-photon pair cannot befactored into a product of two individual quantum states.

In general, more than two photons may be entangled together. More thantwo photons entangled together are referred to as “multiply-entangled”photons. Measuring properties of one or more photons in a set ofmultiply-entangled photons restricts properties of the rest of thephotons in the set by constraining measurement outcomes. As understoodby those of ordinary skill in the art and by way of non-limitingexample, the quantum mechanical state of a set of n>2 multiply-entangledphotons cannot be factored into a product of n separate states. The term“entangled photons” refers to both biphotons and multiply-entangledphotons.

SUMMARY OF THE INVENTION

The present invention provides a novel technique for performingcomputations and processing images using entangled photons. The systemsand methods according to certain embodiments of the present inventiontake advantage of faster processing time made available by artful use ofquantum entanglement properties of light. The techniques presentedherein may be adapted to perform a wide range of computations andprocessing algorithms. Thus, embodiments of the present invention may beused for general optical computing and image processing. No generalizedoptical computing and processing techniques having the adaptability andadvantages of the present invention exist in the prior art.

According to an embodiment of the present invention, an apparatus forprocessing entangled complex signals is presented. The apparatusincludes a source of light. The apparatus also includes a polarizer inoptical communication with the source of light configured to producepolarized light. The apparatus further includes a first apertureconfigured to receive light having a first polarization and producefirst encoded light. The apparatus further includes a second apertureconfigured to receive light having a second polarization and producesecond encoded light. The apparatus further includes at least twoadjacent nonlinear crystals configured to receive the first encodedlight and the second encoded light, the two adjacent nonlinear crystalsbeing separated by a distance. The apparatus further includes acoincidence counter configured to detect coincidences between photons.

According to another embodiment of the present invention, a method forprocessing complex signals is presented. The method includes generatingpolarized light. The method also includes splitting the polarized lightinto a first polarized component spatially separated from a secondpolarized component. The method further includes modulating the firstpolarized component with a first complex signal. The method furtherincludes modulating the second polarized component with a second complexsignal. The method further includes directing the first polarizedcomponent and the second polarized component through at least twoadjacent nonlinear crystals. The method further includes manipulating adistance between the adjacent nonlinear crystals. The method furtherincludes performing at least one coincidence measurement. The methodfurther includes determining at least one parameter associated with atleast one of the first complex signal and the second complex signal.

According to another embodiment of the present invention, a method forprocessing complex signals is presented. The method includes providinglight. The method also includes imposing a first signal on a firstpolarized component of the light to produce first encoded light. Themethod further includes imposing a second signal on a second polarizedcomponent of the light to produce second encoded light. The methodfurther includes transmitting the first encoded light and the secondencoded light through adjacent nonlinear crystals separated by adistance. The method further includes determining properties of one ofthe first signal and the second signal using results of at least onecoincidence measurement.

According to another embodiment of the present invention, a method ofencoding classical information as a quantum state is presented. Themethod includes producing light. The method also includes separating thelight into a first polarized component and a second polarized component.The method further includes modulating the first polarized componentwith a first classical signal to produce first modulated light. Themethod further includes modulating the second polarized component with asecond classical signal to produce second modulated light. The methodfurther includes directing the first modulated light and the secondmodulated light through a first downconverter and a seconddownconverter.

Other exemplary embodiments and advantages of the present invention maybe ascertained by reviewing the present disclosure and the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is further described in the detailed descriptionwhich follows, in reference to the noted plurality of drawings by way ofnon-limiting examples of certain embodiments of the present invention,in which like numerals represent like elements throughout the severalviews of the drawings, and wherein:

FIG. 1 is a schematic diagram of an entangled photon source according toan embodiment of the present invention;

FIG. 2 is a schematic diagram of an entangled photon source havingadjustable crystal separation according to an embodiment of the presentinvention;

FIG. 3 is a schematic diagram of an embodiment of the present invention;and

FIG. 4 is a schematic diagram illustrating certain coordinateconventions used herein for various calculations according to certainembodiments of the present invention.

DETAILED DESCRIPTION

The particulars shown herein are by way of example and for purposes ofillustrative discussion of the embodiments of the present invention onlyand are presented in the cause of providing what is believed to be themost useful and readily understood description of the principles andconceptual aspects of the present invention. In this regard, thedescription taken with the drawings provides a fundamental understandingof the present invention, making apparent to those skilled in the arthow the several forms of the present invention may be embodied inpractice.

I. Introduction

In the last decade, researchers have exerted much effort inunderstanding nontrivial polarization properties of light as describedby the theory of quantum mechanics. In particular, researchers continueto expend considerable efforts toward understanding the quantumproperties of polarization-entangled states generated by type-Ispontaneous parametric down-conversion. Such states may be used to studycertain aspects of quantum information processing. In the opticaldomain, they relate to quantum computing, quantum cryptography and ahost of fourth-order interference effects. See, e.g., E. Knill, R.Laflamme, and G. J. Milburn, Nature 409, 46 (2001) (discussion quantumcomputing), T. Jennewein, et al., Phys. Rev. Lett. 84, 4729 (2000)(discussing quantum cryptography), and C. H. Monken, P. H. SoutoRibeiro, and S. Padua, Phys. Rev. A 57, 3123 (1998) (discussingfourth-order interference effects).

One embodiment of the present invention provides a system for and methodof optically encoding classical complex signals intopolarization-entangled states and subsequently manipulating such statesfor the purpose of quantum information processing. In the encoding step,one polarization component of a laser beam is directed to an aperturecorresponding to a first signal, and another polarization component isdirected to an aperture corresponding to a second signal. The twopolarization components are then recombined and directed to a pair ofnon-linear crystals where they undergo type-I spontaneous parametricdownconversion. When a non-linear crystal is thin, a consequence ofphase-matching conditions in the down-conversion process within thecrystal is that the angular spectrum of the incident beam is transferredto the two-photon state resulting from the down-conversion process. Theimage (angular spectrum) encoded in the first polarization component ofthe pump laser beam is thus transferred to a two-photon state via adown-conversion process.

Subsequent manipulation of the resulting quantum state offers theopportunity to control the transverse profile of the coincidencedetection without disturbing the incident intensity. (Photon detectorscoupled to a coincidence counter are placed so as to detect coincidencesbetween photons originating from the same crystal, whether it be thefirst crystal or the second crystal.) As an example of the type ofprocessing that is available once a signal of interest is encoded, anembodiment of the present invention provides an algorithm fordetermining the amplitude and phase of an unknown state, given fourmeasurements on the non-maximally entangled state containing a knownstate as well as the unknown state. For complex signal analysis, e.g.,complex images, multiple entangled degrees of freedom may be employed.By way of non-limiting example, two spatially separated non-linearcrystals may be used to provide one such degree of freedom, crystalseparation in the direction of propagation, for manipulating thedown-converted two-photon state.

One embodiment of the invention involves encoding complex signals andquantum-mechanically processing the encoded complex signals by way ofpolarization-entangled states resulting from type-I spontaneousparametric down-conversion in a pair of spatially separated non-linear(χ⁽²⁾) crystals. By way of non-limiting example, suchclassical-to-quantum encoding may be done by utilizing a laser source(e.g., UV or blue) polarized at some angle θ with respect to thevertical. The beam impinges on a polarization beam splitter sending thehorizontal components down a path separated spatially from the verticalcomponents. The separated horizontal and vertical pump beam componentsimpinge on separate apertures and are subsequently recombined into asingle beam. The combined beam is then directed to a pair of spatiallyseparated non-linear crystals, which transfer the angular spectrum ofthe incident beam to a two-photon state resulting from thedown-conversion process.

The embodiment is further described in terms of quantum mechanicaltechniques of processing the information contained in the two-photonstate and providing a solution to the problem of amplitude and phaseretrieval for an unknown complex signal. Such a solution provides anon-limiting example of an approach to processing classical complexsignals, capitalizing on potential quantum speed-up available only in aquantum computational environment.

The exemplary embodiment is next described in more detail in two parts.The first part relates to the encoding of classical complex signalinformation in an entangled state. The second part relates to, amongother effective techniques, a methodology for amplitude and phasedetermination for the quantum encoded unknown classical signal asachieved through measurements of the fourth-order correlation function.

II. Encoding Information

FIG. 1 is a schematic diagram of an exemplary entangled photon source100 according to an embodiment of the present invention. In particular,source 100 provides a polarization-entangled state prepared using twonon-linear crystals 110, 120. An entangled two-photon state 150 isemitted along cones 115, 125 generated by type-I down-conversion incrossed, adjacent non-linear crystals 110, 120. Cone 115 consists ofhorizontally-polarized light, while cone 125 consists ofvertically-polarized light. Crystals 110, 120 are configured such that aspacing 130 between them may be controlled (in some instances, thecrystals may contact each-other in direct abutment, as depicted in FIG.1). Pump beam 140 is an approximate plane wave pump beam polarized at anangle θ 160 from vertical 170 and is incident on crossed, adjacent,non-linear crystals 110, 120 cut for type-I down-conversion. In thisarrangement, with the pump beam direction and optical axis of firstcrystal 110 defining the vertical plane, and the optical axis of secondcrystal 120 rotated 90° with respect to the axis of first crystal 110,the pump direction and the optical axis of second crystal 120 formingthe horizontal plane, a polarization-entangled two-photon state 150 isemitted by the pair of crystals 110, 120.

In the arrangement depicted in FIG. 1, if the pump polarization isaligned vertically, down-conversion occurs only in first crystal 110.Similarly, if the pump polarization is aligned horizontally,down-conversion occurs only in second crystal 120. When the pumppolarization is set to θ=45°, down-conversion is equally likely ineither crystal 110, 120, so photons are emitted in the maximallyentangled state, which may be represented as, by way of non-limitingexample: $\begin{matrix}{\left. \psi \right\rangle = {{\frac{1}{\sqrt{2}}\left( {\text{❘}H} \right\rangle\text{❘}H\text{〉}} + {{\delta\mathbb{e}}^{\mathbb{i}\phi}\text{❘}V\text{〉}\left. V \right\rangle{\text{)}.}}}} & (1)\end{matrix}$

In the system represented by Equation (1), the relative phase φ iscontrolled by an ultraviolet quarter wave plate in the pump beam. Thesymbols |V> and |H> represent quantum states of vertically- andhorizontally-polarized components. That down conversion is equallylikely in either crystal 110, 120 is approximately true because the pumpbeam power is somewhat attenuated in the second crystal 120 due toabsorption in the first crystal 110. For pump beam polarization at anarbitrary angle θ≠45°, the non-maximally entangled two-photon state 150that emerges from the two-crystal source 100 may be represented as, byway of non-limiting example: $\begin{matrix}{\left. \psi \right\rangle = {{\frac{1}{\sqrt{1 + \delta^{2}}}\left( {\text{❘}H} \right\rangle\text{❘}H\text{〉}} + {{\delta\mathbb{e}}^{\mathbb{i}\phi}\text{❘}V\text{〉}\left. V \right\rangle{\text{)}.}}}} & (2)\end{matrix}$In the Equation (2), the symbol δ represents Tan(θ) (i.e., δ≡Tan(θ)).The remaining terms are as in Equation (1).

FIG. 2 is a schematic diagram of an entangled photon source according toan exemplary embodiment of the present invention. In particular, FIG. 2depicts an entanglement source 200 using type-I down-conversion incrossed, spatially separated, non-linear crystals 210, 220. Similar tothe embodiment of FIG. 1, an entangled two-photon state 250 is emittedalong cones 215, 225 generated by type-I down-conversion in crossed,adjacent non-linear crystals 210, 220. Cone 215 consists ofhorizontally-polarized light, while cone 225 consists ofvertically-polarized light. The crystals 210, 220 are configured suchthat an inter-crystal distance 230 may be manipulated. In someinstances, crystals 201, 220 may directly abut each-other. As in theembodiment of FIG. 1, the pump beam direction and the optical axis offirst crystal 210 lie in the vertical plane, and the pump beam directionand the optical axis of second crystal 220 lie in the horizontal plane.In another similarity with embodiment of FIG. 1, the embodiment of FIG.2 may be used to produce maximally or non-maximally entangled states byselection of an appropriate angle θ 160 with respect to the vertical270.

FIG. 2 depicts two modifications of the embodiment of FIG. 1 that resultin greater control over the produced polarization-entangled state.First, complex transverse momentum structures 260 (multi-mode) areimposed separately on the horizontal and vertical polarizationcomponents of pump beam 240 before recombining them and sending pumpbeam 240 into the two crystals 210, 220. Second, the relative phase ofthe components of the output state 250 are controlled by a small spatialseparation 230 of the two down-conversion crystals.

FIG. 3 is a schematic diagram of an exemplary embodiment of the presentinvention. In particular, FIG. 3 schematically depicts a system for theproduction of an entangled output state |ψ_(out)> 350, given input pairsof classical complex angular spectra {v_(x), v_(y)}, one or both ofwhich may be unknown. In the embodiment of FIG. 3, a laser produces apump beam 340, which is directed to a polarizer configured to impart aselectable polarization to the beam at an angle of θ360 to the vertical370. By way of non-limiting example, θ may be 45°. Polarized pump beam340 is then directed to polarizing beam splitter 375, which separates itinto vertically polarized component 380 and horizontally polarizedcomponent 385.

One of the components, by way of non-limiting example, horizontallypolarized component 385, passes through an adjustable optical delay 397such as a movable mirror set in order to adjustably impart a delay(e.g., no delay) between the polarized beam components 380, 385. Eachcomponent 380, 385 is then directed to a respective aperture 390, 395.Thus, horizontally polarized component 385 is directed through aperture395, which modulates it with a signal v_(x), and vertically polarizedcomponent 380 is directed through aperture 390, which modulates it witha signal v_(y). The signals are complex, in that their amplitude andphase each carry information. Thus, for each signal, and at any giventime, the two values of the information-carrying parameters may beexpressed as a single complex number. One value may be represented bythe real part of the complex number and another value represented by theimaginary part of the complex number. In this manner, each polarizationcomponent 380, 385 is separately encoded with a complex signal. Secondpolarizing beam splitter 355 then combines the encoded vertically andhorizontally polarized components 380, 385 into single beam 365, whichit directs it to a pair of non-linear crystals 310, 320.

In the embodiment of FIG. 3, second non-linear crystal 320 iscontrollably separated a distance d≧0 ( 330) from the first non-linearcrystal. Vertically polarized component 380 of incident pump beam 340produces a horizontally polarized two-entangled photon state 315 throughtype-I spontaneous parametric down-conversion (SPDC). Because thetwo-entangled-photon state contains frequencies ω_(s), ω_(i), (eachapproximately half the pump frequency), which sum to the frequency ofpump beam 340, second crystal 320 is essentially transparent to thisstate. In second crystal 320, horizontal component 385 of pump beam 340produces a vertically polarized two-entangled photon state 325 throughSPDC. When pump beam 340 is polarized at 45° from vertical 370, SPDC isequally likely to occur in first crystal 310 or in second crystal 320.These two processes have high spatial overlap (coherence) and produce amaximally-entangled state 350. In particular, output state 350 includesquadruples of multiply-entangled photons.

Output state |ψ_(out)>∝|ψ_(v) _(x) ({right arrow over(ρ)}₊)>+e^(iφ(d,z,k))|ψ_(v) _(y) ({right arrow over (ρ)}₊)> (350) is asuperposition of a quantum state whose (2-D) transverse momentumdistribution (angular spectrum) is that of the classical input, v_(x),with a state of (2-D) transverse momentum distribution identical to theinput distribution v_(y). These states are in superposition with arelative phase that is linearly dependent on the crystal separation, d330. For thin crystals, the amplitudes of the two states 315, 325 are ofthe form of a convolution of the (complex) spatial distribution with theFresnel-Huygens kernel (F) for free space propagation of a complexsignal. Thus, if one of the inputs, say v_(x)(V_(x)({right arrow over(ρ)}₊) in the spatial domain), represents, by way of non-limitingexample, a synthetic aperture radar phase history, then in the outputstate 350, one would have the amplitude F*V_(x)({right arrow over (ρ)}₊)superposed with the amplitude F*V_(y)({right arrow over (ρ)}₊), with therelative phase between the two amplitudes depending linearly on d 330.

Summarizing the encoding step, images (angular spectrum) encoded inincident pump beam 340 by apertures 390, 395 are transferred to thetwo-photon state 350 via the down-conversion process. They reside inthis entangled quantum state in superposition with a relative phasedifference that can be controlled by manipulating the crystal separation330. This offers the opportunity to control either or both amplitudes(transverse profiles) and their relative phases without disturbing theincident pump intensity.

The encoded entangled state 350 is then processed and measured to revealcertain desired information. This amounts to implementing a quantumprocessing algorithm. Toward designing such an algorithm for a givenparticular purpose, the present disclosure proceeds to characterize theencoded entangled state in detail.

A general structure of the state produced by the encoding step ispresented here. Though presented in relation to the embodiments of FIGS.3 and 4, this discussion is not limited to such embodiments. Forcomputational convenience, and by way of non-limiting example, assumehorizontal component 385 of pump 340 has an imposed complex signal V_(y)and that vertical component 380 of pump 340 has an imposed complexsignal V_(x). As a result of the down-conversion process, polarizationentanglement is transformed into transverse momentum entanglement in thetwo-photon state 350. If this state 350 is cast into a spatialcoordinate basis, the signals are essentially complex images. Thus, thedown-conversion process in the crossed, separated crystals 310, 320results in a state 350 that may be represented as, by way ofnon-limiting example:|ψ>=|ψ_(y)({right arrow over (ρ)}₊ ,z)>+e^(iφ(d′,{right arrow over (ρ)},z))|ψ_(x)({right arrow over (ρ)}₊,z)>.  (3)

In Equation (3), ({right arrow over (ρ)}₊,z) indicate transverse andlongitudinal spatial positions, respectively.

As discussed below in detail in relation to FIG. 4, photo detectorsoperatively coupled to a coincidence counter are placed at the output ofthe non-linear crystals so as to detect coincidences between photonsoriginating from the same crystal, whether it be the first crystal orthe second crystal. Coincidence count rate (R) is proportional to thefourth-order spatial correlation function derived from this maximallyentangled state. The coincidence count rate therefore obeys aproportionality relationship that may be represented as, by way ofnon-limiting example:R∝G*({right arrow over (ρ)}₊)G({right arrow over (ρ)}₊).  (4)In Equation (4), the term G({right arrow over (ρ)}₊) may be representedas, by way of non-limiting example:G({right arrow over (ρ)}₊)=F*V _(y)({right arrow over (ρ)}₊)+e^(i{right arrow over (α)}·{right arrow over (ρ)}) ⁺ F*V _(x)({rightarrow over (ρ)}₊).  (5)In Equation (5), F represents the Fresnel-Huygens kernel (below, {rightarrow over (α)} is shown to be real and linearly proportional to thecrystal separation distance, d′).

If one of the complex signals, say V_(x)({right arrow over (ρ)}₊), isregarded as unknown, one can choose the other complex signal accordingto the user's needs. By way of non-limiting example, choosing${{V_{y\quad}\left( {\overset{->}{\rho}}_{+} \right)} = {\frac{1}{2\pi}{\mathbb{e}}^{{+ {\mathbb{i}}}{\overset{->}{\alpha} \cdot \overset{->}{\rho_{+}}}}}},$allows determination of the unknown signal's amplitude and phase fromfour coincidence measurements, each with a differing a (which iscontrolled by crystal separation 330). The unknown signal has a formthat may be expressed as, by way of non-limiting example:F*V _(x)({right arrow over (ρ)}₊)≡x({right arrow over (ρ)}₊)+iy({rightarrow over (ρ)}₊).  (6)

Thus, x({right arrow over (ρ)}₊) represents the real part ofF*V_(x)({right arrow over (ρ)}₊) and y({right arrow over (ρ)}₊)represents the imaginary part of F*V_(x)({right arrow over (ρ)}₊).Equation (6) uses the property of the Fresnel-Huygens kernel, which maybe expressed as, by way of non-limiting example: $\begin{matrix}{{F*{V_{y}\left( {\overset{->}{\rho}}_{+} \right)}} = {2{\pi\mathbb{e}}^{{- {\mathbb{i}}}\frac{z}{k}{\overset{->}{\alpha} \cdot \overset{->}{\alpha}}}{{V_{y}\left( {\overset{->}{\rho}}_{+} \right)}.}}} & (7)\end{matrix}$

Thus, the coincidence count rate, containing the unknown signal, may beexpressed as, by way of non-limiting example: $\begin{matrix}{R \propto {1 + {{{x\left( {\overset{->}{\rho}}_{+} \right)} + {{iy}\left( {\overset{->}{\rho}}_{+} \right)}}}^{2} + {{2\left\lbrack {{{x\left( {\overset{->}{\rho}}_{+} \right)}{{Cos}\left( \frac{z\quad\alpha^{2}}{k} \right)}} - {{y\left( {\overset{->}{\rho}}_{+} \right)}{{Sin}\left( \frac{z\quad\alpha^{2}}{k} \right)}}} \right\rbrack}.}}} & (8)\end{matrix}$In particular, photo detectors operatively coupled to a coincidencecounter are placed at the output of the non-linear crystals so as todetect coincidences between photons originating from the same crystal,whether it be the first crystal or the second crystal. The results ofcoincidence measurements with four differing {right arrow over (α)}(achieved by four different crystal separation distances) yields datafor the unknown F*V_(x)({right arrow over (ρ)}₊)≡x({right arrow over(ρ)}₊)+iy({right arrow over (ρ)}₊). Note that, in general and by way ofnon-limiting example:F ^(*) *F=δ.  (9)Equations (8) and (9) imply that the unknown complex signal may berepresented as, by way of non-limiting example:F ^(*)*(x({right arrow over (ρ)}₊)+iy({right arrow over (ρ)}₊))=F ^(*)*F*V _(x)({right arrow over (ρ)}₊)=V _(x)({right arrow over(ρ)}₊).  (10)The next section provides detailed derivations of the general stateshown in Equation 3, the general expression for the fourth ordercoherence function summarized in Equation 5, and the unknown signalexample of Equation 10.III. Derivations and Analysis

In an embodiment of the present invention, entangled photons areproduced using a pair of non-linear crystals. More particularly, such anembodiment produces entangled two-photon flux by way of two type-Inon-linear, χ⁽²⁾, crystals arranged such that their optical axes lie inplanes perpendicular to each other. By way of non-limiting example, suchcrystals may be constructed of beta barium borate (BBO). The pump beamand optic axis of the first crystal define the vertical plane; the opticaxis of the second crystal defines the horizontal plane. If the pumppolarization is vertically polarized, down-conversion occurs only in thefirst crystal. If the pump polarization is horizontally polarized,down-conversion occurs only in the second crystal. By pumping with abeam whose polarization is set at 45°, down-conversion is equally likelyin either crystal. The two down-conversion processes are mutuallycoherent because the pump beam is reasonably assumed to be continuous.Under this assumption, which is appropriate, the pump beam has a longcoherence length. Given the coherence and large spatial overlap of theseprocesses, down-converted photons are created in the maximally entangledstate shown above in Equation (1).

Next, classical image inputs are modulated into quantum entangledstates. To transfer classical angular spectra to a quantum mechanicalentangled state (classical-to-quantum encoding) the classical angularspectra are first (classically) encoded in orthogonal angular momentumstates of a pump beam incident on a parametric down-conversion crystal(e.g., beta-barium borate-BBO). Within the crystal, in a non-linear χ⁽²⁾process, pump photons spontaneously generate pairs of entangled photons,conserving overall momentum. Because of this conservation of overallmomentum, the phase matching of all output modes at the crystal exitface provides, for thin crystals, transfer of the incident pumptransverse momentum distribution to the two-photon output state. Thus,an image (angular spectrum) encoded in an incident pump beam istransferred to the two-photon state via the down-conversion process.Subsequent manipulation of the resulting two-photon state offers theopportunity to control the transverse profile of the coincidencedetection without disturbing the incident intensity.

Towards designing such a manipulation, a Hamiltonian for the multi-modeparametric down-conversion process is calculated presently. Classically,it is understood that a time-varying electric field with Cartesiancomponents, E_(k), k=1, 2, 3, induces a slight redistribution ofelectrons when incident on a polarizable dielectric medium. If themedium has bilinear susceptibility χ_(ijk), the i^(th) component of thepolarization due to fields in the j, k directions is given byP_(i)=χ_(ijk)E_(j)E_(k). This, in turn, makes a contribution to theenergy of the electromagnetic field that may be expressed as, by way ofnon-limiting example: $\begin{matrix}\begin{matrix}{{H_{\quad I} = {{\frac{1}{2}{\int_{V}{{P_{i}\left( {\overset{->}{r},t} \right)}{E_{i}\left( {\overset{->}{r},t} \right)}{\mathbb{d}^{3}x}}}} + {c.c.}}},} \\{= {{\frac{1}{2}{\int_{V}{\chi_{ijk}{E_{i}\left( {\overset{->}{r},t} \right)}{E_{j}\left( {\overset{->}{r},t} \right)}{E_{k}\left( {\overset{->}{r},t} \right)}{\mathbb{d}^{3}x}}}} + {c.c.}}}\end{matrix} & (11)\end{matrix}$In Equation (11), the volume of integration V is that of the nonlinearmedium, the symbol {right arrow over (r)} represents Cartesiancoordinates, c.c. represents the complex conjugate of the term thatprecedes it, and the symbol t represents time. If |Ψ(0)> is the state ofthe field at time t=0 in the interaction picture, then the state at alater time t>0 may, by way of non-limiting example, be given by theunitary (according to Û(t)) evolution: $\begin{matrix}{{\text{❘}{\psi(t)}\text{〉}} = {{{\hat{U}(t)}\text{❘}{\Psi(0)}\text{〉}} = {{\mathbb{e}}^{{- \frac{\mathbb{i}}{h}}{\int_{0}^{t}{{{\hat{H}}_{I}{(t^{\prime})}}{\mathbb{d}t^{\prime}}}}}\text{❘}{\Psi(0)}{\text{〉}.}}}} & (12)\end{matrix}$In Equation (12), the symbol Ĥ_(I) represents a Hamiltonian for themulti-mode parametric down-conversion process. In the present instance,the initial state of the down-converted light is the vacuum state of thesignal and idler beam, which may be denoted as, by way of non-limitingexample:|Ψ(0)>=|0>_(s)|0>_(i).  (13)In equation (13), the symbols |0>_(s) and |0>_(i) represent the groundstates of the signal photons and idler photons, respectively. Firstorder perturbation theory applied to Equation (12) yields an expressionthat may be represented as, by way of non-limiting example:$\begin{matrix}{{\text{❘}{\psi(t)}\text{〉}} = {{\text{❘}0\text{〉}_{s}\text{❘}0\text{〉}_{i}} - {\frac{\mathbb{i}}{h}{\int_{0}^{t}{{{\hat{H}}_{I}\left( t^{\prime} \right)}{\mathbb{d}t^{\prime}}\text{❘}0\text{〉}_{s}\text{❘}0{\text{〉}_{i}.}}}}}} & (14)\end{matrix}$For times short compared to the average interval betweendown-conversions, and for a sufficiently weak pump interaction, thisfirst order result provides a quantitative description of thedown-conversion process in a non-linear χ⁽²⁾ crystal.

It is possible to induce coherence in down-conversion without inducingemission. This occurs for the two crystal configuration when a singlepump beam impinges on the first crystal producing an entangled pair ifthe idler photon from this first down-conversion passes to the secondcrystal and is aligned with the idler from the second crystal'sdown-converted pair. The degree of coherence between the twodown-converted signals can be controlled by varying the amplitude of theidler field reaching the second crystal. In one embodiment of thepresent invention, both the signal and idler photons produced by thefirst crystal are aligned with the signal and idler photons produced bythe second crystal. Additionally, the degree of coherence may bemanipulated by slight variations in the spatial separation, d, of thetwo crystals.

For a pair of thin crystals separated by a small distance d′, theevolution operator decomposes into a sum of terms, governed by twointeractions represented by Hamiltonians Ĥ_(I) ¹,Ĥ_(I) ². The firstcorresponds to entangled pair production on the first crystal, and thesecond corresponds to entangled pair production on the second crystal.These evolution operators may be written, by way of non-limitingexample, in the form: $\begin{matrix}\begin{matrix}{\quad{{{{\hat{U}}_{I}^{\quad 1}\left( {{t + \tau},{t;{{\quad\overset{->}{r}}_{1} = 0}}} \right)} \equiv {\frac{i}{\quad\hslash}{\int_{0}^{\quad t}{{\quad\hat{H}}_{I}^{\quad 1}\left( t^{\quad\prime} \right){\mathbb{d}t^{\quad\prime}}}}}} = {C{\int{{\mathbb{d}{\overset{->}{q}}_{s}}{\int{\mathbb{d}{\overset{->}{q}}_{i}}}}}}}} \\{{{v_{V}\left( {{\overset{->}{q}}_{s} + {\overset{->}{q}}_{i}} \right)}{a_{{\overset{->}{ɛ}}_{s}}^{+}\left( {\overset{->}{q}}_{s} \right)}{a_{{\overset{->}{ɛ}}_{i}}^{+}\left( {\overset{->}{q}}_{i} \right)}} + {{h.c}{\text{.}\quad.}}}\end{matrix} & (15) \\\begin{matrix}{{{{\hat{U}}_{I}^{\quad 2}\left( {{t^{\prime} + \tau},{t^{\prime};\quad{\overset{->}{r}}_{2}}} \right)} \equiv {\frac{i}{\quad\hslash}{\int_{0}^{\quad t^{\prime}}{{\quad\hat{H}}_{I}^{\quad 2}\left( t^{\quad{\prime\prime}} \right){\mathbb{d}t^{\quad{\prime\prime}}}}}}} = {C{\int{{\mathbb{d}{\overset{->}{q}}_{s}}{\int{\mathbb{d}{\overset{->}{q}}_{i}}}}}}} \\{{{{\mathbb{e}}^{- {{\mathbb{i}\phi}{({{\overset{->}{q}}_{s},{\overset{->}{q}}_{i},d^{\prime}})}}}{v_{H}\left( {{\overset{->}{q}}_{s} + {\overset{->}{q}}_{i}} \right)}{a_{{\overset{->}{ɛ}}_{s}}^{+}\left( {\overset{->}{q}}_{s} \right)}{a_{{\overset{->}{ɛ}}_{i}}^{+}\left( {\overset{->}{q}}_{i} \right)}} + {{h.c}\text{.}}}\quad}\end{matrix} & (16)\end{matrix}$In Equations (15) and (16), U_(I) ^(j) is the evolution operator for thepump interaction with the j-th crystal, for j=1,2. In these twoexpressions, τ represents the interaction time, assumed for purposes ofcomputational convenience to be much longer than the coherence time ofthe down-converted light T_(dc), and $t^{\prime} = {t + {\frac{d}{c}.}}$The symbols {right arrow over (q)}_(s) and {right arrow over (q)}_(i)represent the transverse momenta of the signal photons and idlerphotons, respectively, and the term h.c. represents the hermitianconjugate of the term that precedes it. The phase in Equation (16) maybe represented as, by way of non-limiting example: $\begin{matrix}{{\phi\left( {{\overset{->}{q}}_{s},{\overset{->}{q}}_{i},d^{\prime}} \right)} \equiv {\left\lbrack {\sqrt{{\overset{->}{k}}_{s}^{2} - {\overset{->}{q}}_{s}^{2}} + \sqrt{{\overset{->}{k}}_{i}^{2} - {\overset{->}{q}}_{i}^{2}} - \sqrt{{\overset{->}{k}}_{p}^{2} - {{{\overset{->}{q}}_{s} + {\overset{->}{q}}_{i}}}^{2}}} \right\rbrack{d^{\prime}.}}} & \left( {16a} \right)\end{matrix}$In Equation (16a), the terms {right arrow over (k)}_(s) and {right arrowover (k)}_(i) represent the 3-momentum of the signal photons and idlerphotons, respectively.

To arrive at these operators, it is reasonably assumed that the firstcrystal is centered at the origin of coordinates and the second crystalis centered at {right arrow over (r)}₂={circumflex over (z)}d′, i.e., adistance d′ from the origin along the z-axis. The pump beam isreasonably assumed to be collimated, monochromatic, propagating alongthe z-axis, and described as a multimode coherent state|v_({circumflex over (ε)}) _(p) ({right arrow over (q)}_(p))>, wherev_({circumflex over (ε)}) _(p) ({right arrow over (q)}_(p)) is the pumpangular spectrum with transverse momentum {right arrow over (q)}_(p) and{right arrow over (ε)}_(p) the direction of the pump polarization. Theaforementioned assumptions are physically reasonable and embraced forthe purpose of computational convenience. Deviation from theseassumptions is contemplated.

The two crystals are cut for Type I phase matching and have optical axesoriented 90° with respect to each-other. In Equation (15), the pump ispolarized at 45° with respect the vertical plane (as defined by the pumpdirection and the optical axis of the first crystal) and hence isequally likely to down-convert in either crystal. In Type Idown-conversion, signal and idler photons have the same polarization,orthogonal to that of the pump beam. So the vertical (respectively,horizontal) component of the pump, incident along the optical axis ofthe first (respectively, second) crystal, produceshorizontally-polarized (respectively, vertically-polarized) signal andidler photons. The signal and idler photons from the first crystal, bothhorizontally polarized, are aligned with the optical axis of the secondcrystal, but at frequencies for which the second crystal is essentiallytransparent. Hence they are transmitted, with a phase delay due to thecontrolled separation between the two crystals, in a cone that overlapsthat of the vertically-polarized, down-converted pairs produced by thesecond crystal.

IV. Coordinate and Coincidence Counting Details

FIG. 4 is a schematic diagram illustrating coordinate notation as usedherein for certain calculations. In particular, FIG. 4 depicts x-, y-,and z-axes at 401, 402, and 403, respectively. First crystal 410 isseparated from second crystal 420 by distance d′ 430. Pump beam 340,polarized at angle θ460 to the vertical 470, is directed to firstcrystal 410 and second crystal 420. Signal and idler beams 450 fromfirst crystal 410 and second crystal 420 are indicated, as are theavalanche photo diodes (“APDs”) 435, 445, which are used for detection.APD 435 is configured to detect signal photons, while APD 445 isconfigured to detect idler photons. Note that the signal photon andidler photon pairs detected by APDs 435, 445 may originate from eitherfirst crystal 410 or second crystal 420. APDs 435, 445 are coupled tocoincidence counter 455. As discussed in detail below, analyzers may beplaced in front of APDs 435, 445 to select polarization states{θ_(s),θ_(i)} for coincidence analysis.

Under the conventions of FIG. 4, the normal ordered fourth ordercorrelation function derived from coincidence counting at APDs 435, 445may be represented as, by way of non-limiting example:C({right arrow over (r)}_(s),{right arrow over(r)}_(i))∝<Ψ|E_({right arrow over (ε)}) _(s) ⁽⁻⁾({right arrow over(r)}_(s))E_({right arrow over (ε)}) _(i) ⁽⁻⁾({right arrow over(r)}_(i))E_({right arrow over (ε)}) _(i) ⁽⁺⁾({right arrow over(r)}_(i))E_({right arrow over (ε)}) _(s) ⁽⁺⁾({right arrow over(r)}_(s))|Ψ>.  (17)In Equation (17), the term |Ψ> is obtained from Equations (14)-(16). Thesymbols {right arrow over (r)}_(s),{right arrow over (r)}_(i) representdetector locations for detection of the signal photons and idlerphotons, respectively. The positive frequency fields at the detectorsthat appear in Equation (17) may be represented as, by way ofnon-limiting example: $\begin{matrix}{{{E_{{\overset{->}{ɛ}}_{\quad\ell}}^{( + )}\left( {\overset{->}{r}}_{\ell} \right)} = {\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{a_{ɛ_{\ell}}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{j\quad\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}},{j = 1},{2;{\ell \in {\left\{ {s,i} \right\}.}}}} & (18)\end{matrix}$The negative frequency fields at the detectors which appear in Equation(17) may be represented as, by way of non-limiting example:$\begin{matrix}{{{E_{{\overset{->}{ɛ}}_{\quad\ell}}^{( - )}\left( {\overset{->}{r}}_{\ell} \right)} = {\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{{a_{ɛ_{\ell}}}^{+}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{j\quad\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}},{j = 1},{2;{\ell \in {\left\{ {s,i} \right\}.}}}} & (19)\end{matrix}$In Equations (18,19), {right arrow over (r)}_(jt) represents thedistance from the center of the j^(th) crystal to the center of the APDdetector face. More particularly, {right arrow over (r)}_(1s) representsthe distance 451 signal photons originating in first crystal 410 travelto detector 435; {right arrow over (r)}_(1i) represents the distance 452idler photons originating in first crystal 410 travel to detector 435;{right arrow over (r)}_(2s) represents the distance 453 signal photonsoriginating in second crystal 420 travel to detector 445; and {rightarrow over (r)}_(2i) represents the distance 454 idler photonsoriginating in second crystal 420 travel to detector 445. Note that thefollowing definitions apply: {right arrow over (r)}_(1s)≡{right arrowover (ρ)}_(s)+z_(1s){circumflex over (z)}, {right arrow over(r)}_(1i)≡{right arrow over (ρ)}_(i)+z_(1i){circumflex over (z)}, {rightarrow over (r)}_(2s)≡{right arrow over (ρ)}_(s)+z_(2s){circumflex over(z)}, and {right arrow over (r)}_(2i)≡{right arrow over(ρ)}_(i)+z_(2i){circumflex over (z)}, where z_(1s), z_(1i), z_(2s), andZ_(2i) represent the z-axis components of {right arrow over (r)}_(1s),{right arrow over (r)}_(1i), {right arrow over (r)}_(2s), and {rightarrow over (r)}_(2i), respectively, and {right arrow over (ρ)}_(s) 411and {right arrow over (ρ)}_(i) 412 represent the transverse spatialpositions of the detected signal and idler photons, respectively.

When pre-detection polarization analysis is performed by placingpolarizers just prior to the signal and idler detectors 435, 445, thedetected fields decompose into a form that may be represented as, by wayof non-limiting example:E _({right arrow over (ε)}) _(l) ^((±))({right arrow over (r)}_(l))→Sin(θ_(l))E _(H) ^((±))({right arrow over (r)} _(l))+Cos(θ_(l))E_(V) ^((±))({right arrow over (r)} _(l)), lε{s,i}.  (20)Expressed in a different form, Equation (20) may be written as, by wayof non-limiting example: $\begin{matrix}{{{{E_{{\overset{->}{ɛ}}_{\quad\ell}}^{( + )}\left( {\overset{->}{r}}_{\ell} \right)} = {{{{Sin}\left( \theta_{\ell} \right)}{\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{a_{H}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{1\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}}} + {{{Cos}\left( \theta_{\ell} \right)}{\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{a_{V}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{2\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}}}}};{\ell \in \left\{ {s,i} \right\}}},} & (21) \\{{{E_{{\overset{->}{ɛ}}_{\quad\ell}}^{( - )}\left( {\overset{->}{r}}_{\ell} \right)} = {{{{Sin}\left( \theta_{\ell} \right)}{\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{a_{H}^{+}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{2\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}}} + {{{Cos}\left( \theta_{\ell} \right)}{\int{{\mathbb{d}^{2}q_{\ell}^{\prime}}{a_{V}^{+}\left( {\overset{->}{q}}_{\ell}^{\prime} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{\ell}^{\prime} \cdot {\overset{->}{\rho}}_{\ell}} + {r_{2\ell}\sqrt{{\overset{->}{k}}_{\ell}^{2} - {\overset{->}{q}}_{\ell}^{\prime 2}}}})}}}}}}};{\ell \in {\left\{ {s,i} \right\}.}}} & (22)\end{matrix}$In Equations (21) and (22), the field operators obey the commutationrelations └a_(α)({right arrow over (k)}),a_(β)({right arrow over(q)})┘=└a⁺ _(α)({right arrow over (k)}),a⁺ _(β)({right arrow over(q)})┘=0 and └a_(α)({right arrow over (k)}),a⁺ _(β)({right arrow over(q)})┘=δ({right arrow over (k)}−{right arrow over (q)})δ_(αβ), where αand β range over horizontal and vertical polarizations. In this lastexpression, the transmission axis of the polarizer in front of thesignal (respectively, idler) detector makes an angle θ_(s)(respectively, θ_(i)) with respect to the vertical polarization planedefined by the pump beam direction and the optical axis of first crystal410. A slight simplification appears in Equations (21) and (22) becausehorizontally-polarized photons are produced in first crystal 410 andvertically-polarized photons are produced in second crystal 420. The‘crystal index’ j can be set to 1 in the first term of each of theseequations and set to 2 in the second term of each equation, asindicated. The fourth order correlation function of Equation (17) can beexplicitly formed using Equations (21) and (22) and the two-photon statevector from Equation (14). Begin by simplifying the expression:E_({right arrow over (ε)}) _(i) ⁽⁺⁾({right arrow over(r)}_(i))E_({right arrow over (ε)}) _(s) ⁽⁺⁾({right arrow over(r)}_(s))|Ψ>.  (23)Repeated use of the commutation relations above yields, by way ofnon-limiting example: $\begin{matrix}{\left. {{{E_{{\overset{->}{ɛ}}_{\quad i}}^{( + )}\left( {\overset{->}{r}}_{i} \right)}{E_{{\overset{->}{ɛ}}_{\quad s}}^{( + )}\left( {\overset{->}{r}}_{s} \right)}}❘\Psi} \right\rangle = {{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{\int{{\mathbb{d}^{2}q_{i}^{\prime}}{\int{{\mathbb{d}^{2}q_{s}^{\prime}}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{i}^{\prime} \cdot {\overset{->}{\rho}}_{i}}r_{1i}\sqrt{{\overset{->}{k}}_{i}^{2} - {\overset{->}{q}}_{i}^{\prime 2}}})}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{s}^{\prime} \cdot {\overset{->}{\rho}}_{s}}r_{1s}\sqrt{{\overset{->}{k}}_{s}^{2} - {\overset{->}{q}}_{s}^{\prime 2}}})}{\int{{\mathbb{d}^{2}q_{i}}{\int{{\mathbb{d}^{2}q_{s}}{\delta\left( {{\overset{->}{q}}_{i} - {\overset{->}{q}}_{i}^{\prime}} \right)}{\delta\left( {{\overset{->}{q}}_{s} - {\overset{->}{q}}_{s}^{\prime}} \right)}{v_{V}\left( {{\overset{->}{q}}_{i} + {\overset{->}{q}}_{s}} \right)}\left. 0 \right\rangle_{s}\text{}0\text{〉}_{i}}}}}}}}}} + {{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{\int{{\mathbb{d}^{2}q_{i}^{\prime}}{\int{{\mathbb{d}^{2}q_{s}^{\prime}}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{i}^{\prime} \cdot {\overset{->}{\rho}}_{i}}r_{2i}\sqrt{{\overset{->}{k}}_{i}^{2} - {\overset{->}{q}}_{i}^{\prime 2}}})}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{s}^{\prime} \cdot {\overset{->}{\rho}}_{s}} + {r_{2s}\sqrt{{\overset{->}{k}}_{s}^{2} - {\overset{->}{q}}_{s}^{\prime 2}}}})}{\int{{\mathbb{d}^{2}q_{i}}{\int{{\mathbb{d}^{2}q_{s}}{\delta\left( {{\overset{->}{q}}_{i} - {\overset{->}{q}}_{i}^{\prime}} \right)}{\delta\left( {{\overset{->}{q}}_{s} - {\overset{->}{q}}_{s}^{\prime}} \right)}{v_{H}\left( {{\overset{->}{q}}_{i} + {\overset{->}{q}}_{s}} \right)}{\mathbb{e}}^{- {{\mathbb{i}\phi}{({{\overset{->}{q}}_{i},{\overset{->}{q}}_{s},d})}}}\text{❘}0\text{〉}_{s}\text{❘}0{\text{〉}_{i}.}}}}}}}}}}}} & (24)\end{matrix}$Integrating out the delta functions simplifies Equation (24) to, by wayof non-limiting example: $\begin{matrix}{= {{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{\int{{\mathbb{d}^{2}q_{i}}{\int{{\mathbb{d}^{2}q_{s}}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{i} \cdot {\overset{->}{\rho}}_{i}} + {r_{1i}\sqrt{{\overset{->}{k}}_{i}^{2} - {\overset{->}{q}}_{i}^{\prime 2}}}})}}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{s} \cdot}{{\overset{->}{\rho}}_{s}r_{1s}\sqrt{{\overset{->}{k}}_{s}^{2} - {\overset{->}{q}}_{s}^{2}}}})}{v_{V}\left( {{\overset{->}{q}}_{i} + {\overset{->}{q}}_{s}} \right)}\left. 0 \right\rangle_{s}\text{}0\text{〉}_{i}}}}}} + {{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{\int{{\mathbb{d}^{2}q_{i}}{\int{{\mathbb{d}^{2}q_{s}}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{i} \cdot {\overset{->}{\rho}}_{i}}r_{2i}\sqrt{{\overset{->}{k}}_{i}^{2} - {\overset{->}{q}}_{i}^{\prime 2}}})}{\mathbb{e}}^{{\mathbb{i}}({{{\overset{->}{q}}_{s} \cdot {\overset{->}{\rho}}_{s}}r_{2s}\sqrt{{\overset{->}{k}}_{s}^{2} - {\overset{->}{q}}_{s}^{2}}})}{v_{H}\left( {{\overset{->}{q}}_{i} + {\overset{->}{q}}_{s}} \right)}{\mathbb{e}}^{- {{\mathbb{i}\phi}{({{\overset{->}{q}}_{i},{\overset{->}{q}}_{s},d})}}}\text{❘}0\text{〉}_{s}\text{❘}0\text{〉}_{i}}}}}}}} & (25)\end{matrix}$This can be further simplified by implementing the following reasonableassumptions: (i) the magnitudes of the signal and idler transversemomenta are small compared to the longitudinal momenta (paraxialapproximation), |{right arrow over (q)}_(s)|<<|{right arrow over(k)}_(s)|,|{right arrow over (q)}_(i)|<<|{right arrow over (k)}_(i)|;(ii) the signal and idler are degenerate in energy, |{right arrow over(k)}_(s)|=|{right arrow over (k)}_(s)|≡k hence the pump momentum is|{right arrow over (k)}_(p)|≡k_(p)=2k; and (iii) r_(1i)=r_(1s)=z,r_(2i)=r_(2s)=z′. With these simplifying assumptions, expanding thesquare roots and the phase term yields, by way of non-limiting example:$\begin{matrix}{\sqrt{{\overset{->}{k}}^{2} - {\overset{->}{q}}_{\ell}^{2}} \approx {k - {\frac{{\overset{->}{q}}_{\ell}^{2}}{2k}.}}} & (26)\end{matrix}$Equation (16a) yields, by way of non-limiting example: $\begin{matrix}\begin{matrix}{{\phi\left( {{\overset{->}{q}}_{s},{\overset{->}{q}}_{i},d^{\prime}} \right)} \approx {\left\lbrack {k - \frac{{\overset{->}{q}}_{s}^{2}}{2k} + k - \frac{{\overset{->}{q}}_{i}^{2}}{2k} - k_{p} + \frac{{{{\overset{->}{q}}_{s} + {\overset{->}{q}}_{i}}}^{2}}{2k_{p}}} \right\rbrack d^{\prime}}} \\{= {\frac{d^{\prime}}{4k}{\left( {{{{\overset{->}{q}}_{s} + {\overset{->}{q}}_{i}}}^{2} - {2\left( {{\overset{->}{q}}_{s}^{2} + {\overset{->}{q}}_{i}^{2}} \right)}} \right).}}}\end{matrix} & (27)\end{matrix}$An additional simplification can be achieved by transforming coordinatesand momenta according to, by way of non-limiting example:$\begin{matrix}{{{\overset{->}{q}}_{\pm} \equiv {\frac{1}{2}\left( {{\overset{->}{q}}_{i} \pm {\overset{->}{q}}_{s}} \right)}},{{\overset{->}{\rho}}_{\pm} \equiv {\left( {{\overset{->}{\rho}}_{i} \pm {\overset{->}{\rho}}_{s}} \right).}}} & (28)\end{matrix}$Equation (28) may be used to transform Equation (25) into, by way ofnon-limiting example: $\begin{matrix}{= {{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{\mathbb{e}}^{2{{\mathbb{i}}{kz}}}{\int{{\mathbb{d}^{2}q_{+}}{v_{V}\left( {\overset{->}{q}}_{+} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{+} \cdot {\overset{->}{\rho}}_{+}} - \frac{{\overset{->}{q}}_{+}^{2}z}{k}})}}{\int{{\mathbb{d}^{2}{q\_\mathbb{e}}^{({{{\overset{->}{q}}_{-} \cdot {\overset{->}{\rho}}_{-}} - \frac{{\overset{->}{q}}_{-}^{2}z}{k}})}}\text{❘}0\text{〉}_{s}\text{❘}0\text{〉}_{i}}}}}} + {{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{\mathbb{e}}^{2{\mathbb{i}}\quad{kz}^{\prime}}{\mathbb{e}}^{- {{\mathbb{i}}(\frac{{\overset{->}{\rho}}_{+} \cdot \overset{->}{d^{\prime}}}{z^{\prime}})}}{\int{{\mathbb{d}^{2}q_{+}}{v_{H}\left( {\overset{->}{q}}_{+} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{+} \cdot {\overset{->}{\rho}}_{+}} - \frac{{\overset{->}{q}}_{+}^{2}z^{\prime}}{k}})}}{\int{{\mathbb{d}^{2}q_{-}}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{-} \cdot {\overset{->}{\rho}}_{-}} - \frac{{\overset{->}{q}}_{-}^{2}{({z^{\prime} - d^{\prime}})}}{k}})}}\text{❘}0\text{〉}_{s}\text{❘}0{\text{〉}_{i}.}}}}}}}} & (29)\end{matrix}$In Equation (29), {right arrow over (d)}′={circumflex over (z)}d′. Thed²q⁻ integrals can be performed using, by way of non-limiting example,${\int{{\mathbb{d}^{2}q}\quad{\mathbb{e}}^{{\mathbb{i}}({{\overset{->}{q} \cdot \overset{->}{\rho}}{\, -^{\underset{\_}{{\overset{->}{q}}^{2}a}}}})}}} = {{\mathbb{e}}^{{\mathbb{i}}\frac{\rho^{2}}{4a}}.}$

The remaining integrations may be defined as, by way of non-limitingexample: $\begin{matrix}{{{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)} \equiv {\int{{\mathbb{d}^{2}q_{+}}{v_{V}\left( {\overset{->}{q}}_{+} \right)}{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{+} \cdot {\overset{->}{\rho}}_{+}} - \frac{{\overset{->}{q}}_{+}^{2}z}{k}})}}}}},{{W_{H}\left( {{\overset{->}{\rho}}_{+},z^{\prime}} \right)} \equiv {\int{{\mathbb{d}^{2}q_{+}}{v_{H}\left( {\overset{->}{q}}_{+} \right)}{{\mathbb{e}}^{{\mathbb{i}}{({{{\overset{->}{q}}_{+} \cdot {\overset{->}{\rho}}_{+}} - \frac{{\overset{->}{q}}_{+}^{2}z^{\prime}}{k}})}}.}}}}} & (30)\end{matrix}$Setting the detectors at z=z′ for purposes of computational expediency,Equation (29) takes a form (up to an overall phase) that may berepresented as, by way of non-limiting example: $\begin{matrix}{\left. {{{E_{\overset{->}{ɛ_{i}}}^{( + )}\left( {\overset{->}{r}}_{i} \right)}{E_{\overset{->}{ɛ_{s}}}^{( + )}\left( {\overset{->}{r}}_{s} \right)}}❘\Psi} \right\rangle = {\begin{pmatrix}{{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{\mathbb{e}}^{{\mathbb{i}}{\overset{->}{\rho}}_{-}^{2}\frac{k}{4z}}{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}} +} \\{{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{\mathbb{e}}^{- {{\mathbb{i}}(\frac{k{({{\overset{->}{\rho}}_{+} \cdot {\overset{->}{d}}^{\prime}})}}{z})}}{\mathbb{e}}^{{\mathbb{i}}{\overset{->}{\rho}}_{-}^{2}\frac{k}{4{({z - d^{\prime}})}}}{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}}\end{pmatrix}\text{❘}0\text{〉}_{s}\text{❘}0{\text{〉}_{i}.}}} & (31)\end{matrix}$An additional overall phase,${\mathbb{e}}^{{\mathbb{i}}{\overset{->}{\rho}}_{-}^{2}\frac{k}{\quad{4z}}},$can be removed, yielding, by way of non-limiting example:$\begin{matrix}{\left. {{{E_{\overset{->}{ɛ_{i}}}^{( + )}\left( {\overset{->}{r}}_{i} \right)}{E_{\overset{->}{ɛ_{s}}}^{( + )}\left( {\overset{->}{r}}_{s} \right)}}❘\Psi} \right\rangle = {\begin{pmatrix}{{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}} +} \\{{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}} \\{{\mathbb{e}}^{- {{\mathbb{i}}(\frac{k{({{\overset{->}{\rho}}_{+} \cdot {\overset{->}{d}}^{\prime}})}}{z})}}{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}}\end{pmatrix}\text{❘}0\text{〉}_{s}\text{❘}0{\text{〉}_{i}.}}} & (32)\end{matrix}$In Equation (32), terms of relative order $\begin{matrix}\frac{{\overset{->}{\rho}}_{-}}{z} & \quad\end{matrix}$have been dropped in the exponential. Dropping such terms simplifiescalculations without adversely affecting the analysis provided by theequations. The fourth order correlation, corresponding to coincidencedetections between the signal and idler detectors, is obtained usingthis last expression and its hermitian conjugate in Equation (17). Thiscorrelation function may then be represented as, by way of non-limitingexample: $\begin{matrix}{{C\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\theta_{i}},\theta_{s}} \right)} = {{c\begin{pmatrix}\begin{matrix}{{{{Sin}^{2}\left( \theta_{i} \right)}{{Sin}^{2}\left( \theta_{s} \right)}{{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}}^{2}} +} \\{{{{Cos}^{2}\left( \theta_{i} \right)}{{Cos}^{2}\left( \theta_{s} \right)}{{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}}^{2}} +}\end{matrix} \\{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)} \times} \\\begin{bmatrix}{{{\mathbb{e}}^{\mathbb{i}ɛ}{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}{W_{H}^{*}\left( {{\overset{->}{\rho}}_{+},z} \right)}} +} \\{{\mathbb{e}}^{- {\mathbb{i}ɛ}}{W_{V}^{*}\left( {{\overset{->}{\rho}}_{+},z} \right)}{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}}\end{bmatrix}\end{pmatrix}}.}} & (33)\end{matrix}$In Equation (33),${ɛ \equiv \frac{- {k\left( {{\overset{->}{\rho}}_{+} \cdot {\overset{->}{d}}^{\prime}} \right)}}{z}},$and c is an overall constant. In the following discussion, ({right arrowover (ρ)}₊,z) will be assumed to be fixed.

A more suggestive, and in some respects more convenient, notationincorporating this last expression can be developed by defining thehermitian coherency matrix as, by way of non-limiting example:$\begin{matrix}\begin{matrix}{{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z} \right)} = \begin{pmatrix}{{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}}^{2} & {{\mathbb{e}}^{- {\mathbb{i}ɛ}}{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)}{W_{V}^{*}\left( {{\overset{->}{\rho}}_{+},z} \right)}} \\{{\mathbb{e}}^{\mathbb{i}ɛ}{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}{W_{H}^{*}\left( {{\overset{->}{\rho}}_{+},z} \right)}} & {{W_{V}\left( {{\overset{->}{\rho}}_{+},z} \right)}}^{2}\end{pmatrix}} \\{\equiv {\begin{pmatrix}{J_{HH}\left( {{\overset{->}{\rho}}_{+},z} \right)} & {{J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)}{\mathbb{e}}^{- {\mathbb{i}ɛ}}} \\{{J_{VH}\left( {{\overset{->}{\rho}}_{+},z} \right)}{\mathbb{e}}^{\mathbb{i}ɛ}} & {J_{VV}\left( {{\overset{->}{\rho}}_{+},z} \right)}\end{pmatrix}.}}\end{matrix} & (34)\end{matrix}$Toward such a convenient notation, define the complex unit vector as, byway of non-limiting example: $\begin{matrix}{{{\chi\left( {\theta_{i},\theta_{s}} \right)} \equiv {\frac{1}{N\left( {\theta_{i},\theta_{s}} \right)}\begin{pmatrix}{{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}} \\{{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}}\end{pmatrix}}},{{N^{2}\left( {\theta_{i},\theta_{s}} \right)} = {\left( {{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}} \right)^{2} + {\left( {{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}} \right)^{2}.}}}} & (35)\end{matrix}$Using these two definitions and absorbing the overall constant c into C,a ‘normalized’ coincidence count rate can obtained from Equation (33) inthe form of a trace that may be represented as, by way of non-limitingexample: $\begin{matrix}\begin{matrix}{{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\theta_{i}},\theta_{s}} \right)} \equiv \frac{C\left( {\overset{->}{\rho},z,{ɛ;},\theta_{i},\theta_{s}} \right)}{N^{2}\left( {\theta_{i},\theta_{s}} \right)}} \\{= {{{Tr}\left\lbrack {{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z,ɛ} \right)}{\rho\left( {\theta_{i},\theta_{s}} \right)}} \right\rbrack}.}}\end{matrix} & (36)\end{matrix}$In Equation (36), ρ(θ_(i),θ_(s)) is a density matrix, defined as, by wayof non-limiting example: $\begin{matrix}\begin{matrix}{{\rho\left( {\theta_{i},\theta_{s}} \right)} \equiv {{\chi\left( {\theta_{i},\theta_{s}} \right)}{\chi^{+}\left( {\theta_{i},\theta_{s}} \right)}}} \\{= \begin{pmatrix}{{{Cos}^{2}\left( \theta_{i} \right)}{{Cos}^{2}\left( \theta_{s} \right)}} & {{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}} \\{{{Cos}\left( \theta_{i} \right)}{{Cos}\left( \theta_{s} \right)}{{Sin}\left( \theta_{i} \right)}{{Sin}\left( \theta_{s} \right)}} & {{{Sin}^{2}\left( \theta_{i} \right)}{{Sin}^{2}\left( \theta_{s} \right)}}\end{pmatrix}}\end{matrix} & (37)\end{matrix}$In equation (37), χ⁺ designates the hermitian conjugate of χ. Using thisnotation, measurements are completely described by specifying the set{ε,ρ(θ_(i),θ_(s))}. This is shown to be the case by noting the followingmeasurement set serves to determine the full coherency matrix

: $\begin{matrix}{\left( {ɛ,{\rho\left( {0,0} \right)}} \right\},\left\{ {ɛ,{\rho\left( {\frac{\pi}{2},\frac{\pi}{2}} \right)}} \right\},\left\{ {0,{\rho\left( {\frac{\pi}{4},\frac{\pi}{4}} \right)}} \right\},{\left\{ {\frac{\pi}{2},{\rho\left( {\frac{\pi}{4},\frac{\pi}{4}} \right)}} \right\}.}} & (38)\end{matrix}$The required relationships, based entirely on measured coincidence dataand hermiticity of the off-diagonal elements, may be expressed as, byway of non-limiting example: $\begin{matrix}\begin{matrix}{{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;0},0} \right)} = {{Tr}\left\lbrack {{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z,ɛ} \right)}{\rho\left( {0,0} \right)}} \right\rbrack}} \\{{= {J_{HH}\left( {{\overset{->}{\rho}}_{+},z} \right)}},}\end{matrix} & (39) \\\begin{matrix}{{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\frac{\pi}{2}},\frac{\pi}{2}} \right)} = {{Tr}\left\lbrack {{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z,ɛ} \right)}{\rho\left( {\frac{\pi}{2},\frac{\pi}{2}} \right)}} \right\rbrack}} \\{{= {J_{VV}\left( {{\overset{->}{\rho}}_{+},z} \right)}},}\end{matrix} & (40) \\\begin{matrix}{{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{0;\frac{\pi}{4}},\frac{\pi}{4}} \right)} = {{Tr}\left\lbrack {{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z,0} \right)}{\rho\left( {\frac{\pi}{4},\frac{\pi}{4}} \right)}} \right\rbrack}} \\{= {{\frac{1}{4}\left( {{J_{HH}\left( {{\overset{->}{\rho}}_{+},z} \right)} + {J_{VV}\left( {{\overset{->}{\rho}}_{+},z} \right)}} \right)} +}} \\{{\frac{1}{\quad 2}{{Re}\left\lbrack {J_{\quad{HV}}\quad\left( \quad{{\overset{\quad->}{\rho}}_{+},z} \right)} \right\rbrack}},}\end{matrix} & (41) \\\begin{matrix}{{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{\frac{\pi}{2};\frac{\pi}{4}},\frac{\pi}{4}} \right)} = {{Tr}\left\lbrack {{\overset{\leftrightarrow}{J}\left( {{\overset{->}{\rho}}_{+},z,\frac{\pi}{2}} \right)}{\rho\left( {\frac{\pi}{4},\frac{\pi}{4}} \right)}} \right\rbrack}} \\{= {{\frac{1}{4}\left( {{J_{HH}\left( {{\overset{->}{\rho}}_{+},z} \right)} + {J_{VV}\left( {{\overset{->}{\rho}}_{+},z} \right)}} \right)} +}} \\{\frac{1}{2}{{{Im}\left\lbrack {J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} \right\rbrack}.}}\end{matrix} & (42)\end{matrix}$Equations (39) and (40) serve to define the diagonal elements of thecoherency matrix. Given these two measurements, Equations (41) and (42)can be seen to yield (complex) off-diagonal terms that may berepresented as, by way of non-limiting example: $\begin{matrix}{{J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} \equiv {{{Re}\left\lbrack {J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} \right\rbrack} + {{{\mathbb{i}Im}\left\lbrack {J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} \right\rbrack}2{{Re}\left\lbrack {J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} \right\rbrack}4{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{0;\frac{\pi}{4}},\frac{\pi}{4}} \right)}} - {\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\frac{\pi}{2}},\frac{\pi}{2}} \right)} - {{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;0},0} \right)}.}}} & \left( {43a} \right) \\{2{{Im}\left\lbrack {{J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)} = {{4\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{\frac{\pi}{2};\frac{\pi}{4}},\frac{\pi}{4}} \right)} - {\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\frac{\pi}{2}},\frac{\pi}{2}} \right)} - {{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,0,0} \right)}.}}} \right.}} & \left( {43b} \right)\end{matrix}$From these last expressions, the remaining off-diagonal terms may berepresented as, by way of on-limiting example:J _(HV)({right arrow over (ρ)}₊ ,z)=J _(VH) ⁺({right arrow over (ρ)}₊,z).  (44)

To summarize to this point, an input pump beam consisting of two images,one horizontally polarized and one vertically polarized, arepolarization entangled by passage through two separated, non-linearcrystals 410, 420 cut for type-I down-conversion with optical axes atright angles to each-other. The pump 440 incidence direction and theoptical axis of the first crystal 410 determine a ‘plane ofpolarization’ for the entire system. Through the phase matchingrequirements of the down-conversion process, polarization entanglementis converted to transverse momentum entanglement for the two images andfor any subset of spatial positions (e.g., pixels) within each of theentangled images. The resulting quantum state consists of asuperposition of two (propagated) complex images whose transversemomenta are entangled.

Users then perform two-photon coincidence counting. This results inmeasurements with a spatial profile consisting of a mix of polarizationstates. The mixture content is determined by pre-detection polarizationanalysis performed at coincidence detectors 435, 445, 455. The objectdescribing the degree of coherence of the entangled images, the complexcoherency matrix, has been related to coincidence measurement data. Aset of four measurements determines the full complex coherency matrix atthe spatial location defined by coincidence detector placement. By usinga paired, calibrated array of coincidence detectors (by way ofnon-limiting example), the complex coherency matrix may be determined asa function of ({right arrow over (ρ)}₊,z).

Determination of the amplitude and phase of an unknown signal, sayW_(H)({right arrow over (ρ)}₊,z), can be achieved by encoding theunknown v_(H)({right arrow over (q)}₊) transverse momentum mask on thehorizontally polarized input and encoding the following expression onthe vertically polarized input: $\begin{matrix}{{v_{V}\left( {\overset{->}{q}}_{+} \right)} = {\frac{A}{2\pi}{\mathbb{e}}^{{- {\mathbb{i}}}\quad{\overset{->}{q}}_{+}^{2}\frac{z}{k}}{{\delta\left( {\overset{->}{q}}_{+} \right)}.}}} & (45)\end{matrix}$In Equation (45), A is a constant. Combining the form of v_(V)({rightarrow over (q)}₊) from Equation (45) with Equation (30) yieldsW_(V)({right arrow over (ρ)}₊,z)=A. The elements of the coherency matrixfor these inputs may be represented as, by way of non-limiting example:J _(HH)({right arrow over (ρ)}₊ ,z)=|W _(H)({right arrow over (ρ)}₊,z)|² , J _(VV)({right arrow over (ρ)}₊ ,z)=A ²,J _(HV)({right arrow over (ρ)}₊ ,z)=A(Re[W _(H)({right arrow over (ρ)}₊,z)]+iIm[W _(H)({right arrow over (ρ)}₊ ,z)]), J _(VH)({right arrow over(ρ)}₊ ,z)=J* _(HV)({right arrow over (ρ)}₊ ,z).  (46)Equation(46) yields, for the unknown complex signal and by way ofnon-limiting example: $\begin{matrix}{{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)} = {\frac{1}{\sqrt{J_{VV}}}{{J_{HV}\left( {{\overset{->}{\rho}}_{+},z} \right)}.}}} & (47)\end{matrix}$In terms of the measurements from Equations (39)-(42) and Equations(43)-(44), this last expression for the unknown signal may be expressedas, by way of non-limiting example: $\begin{matrix}{{W_{H}\left( {{\overset{->}{\rho}}_{+},z} \right)} = {\frac{1}{\sqrt{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\frac{\pi}{2}},\frac{\pi}{2}} \right)}}{\begin{pmatrix}{2\left( {{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{0;\frac{\pi}{4}},\frac{\pi}{4}} \right)} + {i\quad{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{\frac{\pi}{2};\frac{\pi}{4}},\frac{\pi}{4}} \right)}}} \right)} \\{{- \frac{1 + i}{2}}\left( {{\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;\frac{\pi}{2}},\frac{\pi}{2}} \right)} + {\hat{C}\left( {{\overset{->}{\rho}}_{+},z,{ɛ;0},0} \right)}} \right)}\end{pmatrix}.}}} & (48)\end{matrix}$

According to certain embodiments of the present invention, variousdegrees of freedom may be exploited. By way of non-limiting example, thefollowing parameters may be manipulated: polarization states (e.g.,θ_(s), θ_(i)), transverse spatial positions (e.g., {right arrow over(ρ)}_(s), {right arrow over (ρ)}_(i), {right arrow over (ρ)}₊, {rightarrow over (ρ)}₃₁ ), longitudinal spatial position (e.g., z), crystalseparation (e.g., d), and delay. Any of these parameters may be adjustedduring operation of an embodiment of the present invention and/or usedas independent variables, e.g., to facilitate solving systems ofequations.

V. Additional Features and Modifications

Certain embodiments of the present invention may be viewed as generalpurpose quantum optical computers. In such embodiments, a firstpolarized light component is encoded with first information, theproperties of which may be regarded as completely or partially unknown.A second polarized light component is encoded with information that maybe analogized to a computer program, which is used to process or gatherinformation regarding the first information. (In the embodiment of FIG.3, the first information corresponds to signal v_(y), which is encodedinto vertically polarized component 380. The second informationcorresponds to signal v_(x), which is encoded into horizontallypolarized component 385.) Thus, such embodiments are “programmed” byencoding second information (e.g., v_(x) of FIG. 3) onto a polarizedlight component (e.g., 385 of FIG. 3). This “programming” allows thegeneral-purpose quantum optical computer to process or determineproperties of the first information encoded in the first polarized lightcomponent. Such embodiments conduct and analyze coincidence countmeasurements for different values of any, or a combination, ofparameters such as polarization, transverse or longitudinal spatialpositions, crystal separation, and delay. Analyzing the measuredcoincidence counts in accordance with the equations and supportingdiscussions presented herein reveals information regarding the firstsignal as selected by choice of “program” encoded in the second signal.

According to certain embodiments of the present invention, sensors maybe configured to derive information regarding a collection of entangledimage portions (e.g., pixels or qbits) within the image data. Suchembodiments may employ two detectors: a first detector used to detectphotons in multiple locations, and a second detector used to detectphotons in a single location corresponding to a pixel. Because the firstdetector may be configured to globally detect photons, it need not bemoved during operation of the embodiment. The second detector, on theother hand, may be configured to detect individual photons that areentangled with photons detected by the first detector. By moving thesecond detector to multiple locations and performing signal processingat each location, an embodiment of the present invention may be used toderive information about each of a plurality of pixels. Alternately, twostationary detectors capable of detecting photons in multiple locationsmay be used. That is, a bank of stationary detectors may be used insteadof a movable detector. Such a bank may be paired with a detector capableof detecting photons at multiple locations.

Each constituent photon of an entangled photon pair has a smallneighborhood about it (a “correlation neighborhood”) in which spatiallocations are highly correlated. Spatial locations or positionsseparated by distances greater than the associated correlation lengthare spatially independent and may be considered to be independentpixels. Thus, each correlation neighborhood about a particular spatiallocation may be considered as a separate pixel.

Entangled pixel pairs may be considered as qbits. Elemental spatialpositions (e.g., pixels) are in an entangled state. That is, each pixelfrom a first image (e.g., V_(x)) is entangled with a corresponding pixelfrom a second image (e.g., V_(y)), thus forming a single-qbit state. Anyplane passing through entangled image-encoded beams according to certainembodiments of the present invention will contain such qbits.Accordingly, embodiments of the present invention may be used toconstruct a two-dimensional presentation of qbits. In general,spatially-distinct subsets of such a presentation may be considered andprocessed independently (e.g., with multiple detectors).

A two-dimensional presentation of qbits may also contain entangledmulti-qbit states, because any given collection of qbits in thepresentation may be entangled together. In certain embodiments of thepresent invention, each pixel in a given subset of pixels from a firstimage is entangled with a corresponding pixel from a second image. Sucha collection of entangled pixel pairs forms a multi-qbit state. Acollection of qbits considered as an entangled multi-qbit state may ormay not form a spatially continuous region. For example, a two-qbitentangled state may exist as two spatially separate regions in apresentation.

According to certain embodiments of the present invention, non-localoperations and superposition may be used to manipulate quantum states.Unlike classical signal processing, which exploits local correlationsand measurements, processing in a quantum environment is not restrictedto local operations. One type of non-local processing is teleportation.(A measurement is “non-local” if it cannot be reduced to a finitesequence of local measurements.) While teleportation is generally notconsidered a computational resource in any known algorithm, it doesillustrate the fundamental utility of entanglement and non-localprocessing (measurement). Another feature that may be exploited indeveloping quantum algorithms for processing classical signals encodedin the quantum states described herein is superposition.

Entangled photons may be produced from photons of any frequencyconsistent with the present invention.

Conventional computer equipment and applications may be used toaccomplish the various calculations, comparisons, and judgments requiredduring operation according to embodiments of the present invention. Suchcalculations, comparisons, and judgments are preferably performedautomatically during the normal course of operation of embodiments ofthe present invention. Exemplary calculations, comparisons, andjudgments that may be so performed include those elucidated by theenumerated Equations disclosed herein.

Note that the terms “signal” and “idler” may be used interchangeably.More particularly, as used herein, no distinction is drawn betweensignal photons and idler photons. The term “modulate” may be used todescribe encoding information as discussed herein.

Other types of entangled photons and techniques for producing themwithin the scope of the present invention include the following. Thoseof ordinary skill in the art are capable of producing entangled-photonpairs, triples, etc. By way of non-limiting example, entangled photonsmay be produced according to types I or II parametric down-conversion.That is, biphotons whose constituent signal and idler photons areorthogonally polarized may be used as well as biphotons whoseconstituent signal and idler photons are polarized in parallel. Fortype-I down-conversion, signal photons may be separated from idlerphotons (and recombined with idler photons) using dichroic glass. Forboth types of down-conversion, signal photons and idler photos may beselected as they exit the biphoton source by providing apertures at theappropriate angles. Any non-central symmetric nonlinear crystal, notlimited to BBO, may be used. Other ways to produce entangled photonsinclude: excited gasses, materials without inversion symmetry, andgenerally any properly phase-matched medium. Furthermore, the entangledphotons are not limited to any particular wavelength or frequency.

In embodiments of the present invention, coincidence counting may beaccomplished according to a variety of forms. By way of non-limitingexample, avalanche photodiodes, photo multiplier tubes, or other devicesmay be used to detect photons consistent with the present invention.That is, the invention is not limited to the use of avalanchephotodiodes.

The equations used to describe the exemplary embodiments of theinvention described herein are illustrative and representative and arenot meant to be limiting. Alternate equations may be used to representthe same phenomena described by any given equation disclosed herein. Inparticular, the equations disclosed herein may be modified by adding orremoving error-correction terms, higher-order terms, or otherwiseaccounting for inaccuracies, using different names for constants orvariables, using different expressions, or accounting for propagation oflight through different media. Other modifications, substitutions,replacements, or alterations of the equations may be performed inkeeping with the present invention.

In certain portions of this disclosure, certain assumptions are made forthe purpose of computational convenience. Such assumptions are generallyjustified because the situations that they describe are physicallypracticable. Such assumptions also generally yield equations thatprovide robust approximations to the behavior of the actual physicalsystems under consideration. Moreover, the equations and analysisassociated with such assumptions may be modified by those of skill inthe art to accommodate reasonable deviations from the assumptions, ascontemplated by the inventor. That is, embodiments of the presentinvention may depart from these assumptions. Those of ordinary skill inthe art may adjust the analysis presented herein, if necessary, forembodiments that depart from such assumptions.

The particular optical manipulation devices depicted herein areillustrative and representative and are not meant to be limiting. By wayof non-limiting example, apertures, filters, lenses, and particularlasers disclosed herein may be replaced with devices known to those ofordinary skill in the art.

Note that this disclosure follows standard physics notation andnotational conventions. By way of non-limiting example, in some placesPlanck's constant

(and h) and the speed of light c may both considered to be one (1) forthe purpose of calculations. This convention allows, inter alia, forcommon units for frequency and energy, as well as common units for timeand distance (e.g., temporal delays may be considered as spatial lengthsand vice versa). This notational convention is accounted for aftercalculations have been performed in order to deduce correct units forapplication purposes. This disclosure also uses standard physicalnotations known to those of ordinary skill in the art, such as Diracbracket notation (e.g., |ψ_(i)>) to denote quantum states, i to denotethe square root of negative one, and e to denote the natural logarithmbase.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to limit the scope of the presentinvention. As used throughout this disclosure, the singular forms “a,”“an,” and “the” include plural reference unless the context clearlydictates otherwise.

It is noted that the foregoing examples have been provided merely forthe purpose of explanation and are in no way to be construed as limitingof the present invention. While the present invention has been describedwith reference to certain embodiments, it is understood that the wordswhich have been used herein are words of description and illustration,rather than words of limitation. Changes may be made without departingfrom the scope and spirit of the present invention in its aspects.Although the present invention has been described herein with referenceto particular means, materials and embodiments, the present invention isnot intended to be limited to the particulars disclosed herein; rather,the present invention extends to all functionally equivalent structures,methods and uses.

1. An apparatus for processing complex signals, the apparatuscomprising: a source of light; a polarizer in optical communication withthe source of light, the polarizer configured to produce polarizedlight; a polarizing beam splitter in optical communication with thepolarizer, the polarizing beam splitter configured to produce lighthaving a first polarization and light having a second polarization, thefirst polarization being different from the second polarization; a firstaperture configured to receive the light having the first polarizationand produce first encoded light, the first encoded light being encodedwith first information; a second aperture configured to receive thelight having the second polarization and produce second encoded light,the second encoded light being encoded with second information; at leasttwo adjacent nonlinear crystals configured to receive the first encodedlight and the second encoded light, the two adjacent nonlinear crystalsbeing separated by a distance; and a coincidence counter configured todetect coincidences between entangled photons, the entangled photonsexiting the at least two adjacent nonlinear crystals.
 2. The apparatusof claim 1 where the at least two nonlinear crystals comprise a firstcrystal having a first optical axis and a second crystal having a secondoptical axis, wherein the first optical axis is perpendicular to thesecond optical axis.
 3. The apparatus of claim 1 further comprising anoptical delay interposed between the first aperture and the at least twoadjacent nonlinear crystals.
 4. The apparatus of claim 1 wherein thepolarizing beam splitter has a first polarization direction, wherein thepolarized light has a second polarization direction, and wherein thefirst polarization direction is at an angle of 45° with respect to thesecond polarization direction.
 5. The apparatus of claim 1 furthercomprising means for detecting photons at a plurality of locations, themeans for detecting photons in operative communication with thecoincidence counter.
 6. The apparatus of claim 1 further comprisinglogic configured to calculate a parameter associated with the firstinformation and the second information.
 7. The apparatus of claim 6wherein the parameter comprises at least part of a complex coherencymatrix.
 8. The apparatus of claim 6 wherein the first information issubstantially unknown and the parameter comprises information selectedfrom the group consisting of: an amplitude of an unknown signalassociated with the first information, and a phase of an unknown signalassociated with the first information.
 9. The apparatus of claim 6,wherein the first information at least partially determines theparameter.
 10. A method for processing complex signals, the methodcomprising: generating polarized light; splitting the polarized lightinto a first polarized component spatially separated from a secondpolarized component, the first polarized component and the secondpolarized component collectively comprising first entangled photons;modulating the first polarized component with a first signal; modulatingthe second polarized component with a second signal; directing the firstpolarized component and the second polarized component through at leasttwo adjacent nonlinear crystals to produce light comprising secondentangled photons; performing a plurality of coincidence measurements onthe light comprising second entangled photons; and determining, based onthe plurality of coincidence measurements, at least one parameterassociated with the first signal.
 11. The method of claim 10, whereinthe at least two adjacent nonlinear crystals are configured to havesubstantially perpendicular optical axes.
 12. The method of claim 10further comprising delaying the first polarized component.
 13. Themethod of claim 10 wherein the step of splitting comprises passing thepolarized light through a polarizing beam splitter, wherein apolarization direction of the polarized light is at an angle of 45° to apolarization direction of the polarizing beam splitter.
 14. The methodof claim 10 wherein the step of performing a plurality of coincidencemeasurements comprises performing a plurality of coincidencemeasurements at different locations.
 15. The method of claim 10 whereinthe step of performing a plurality of coincidence measurements comprisesperforming a plurality of coincidence measurements for differentdistances between the at least two adjacent nonlinear crystals.
 16. Themethod of claim 10 further comprising transferring an angular spectrumassociated with the first signal and the second signal to an entangledphoton state.
 17. The method of claim 10 wherein the parameter isselected from the group consisting of: an amplitude of the first signal,a phase of the first signal, an amplitude of the second signal, a phaseof the second signal, and a complex coherency matrix associated with thefirst signal and the second signal.
 18. The method of claim 10 whereinthe steps of performing and determining comprise determining afourth-order correlation function.
 19. The method of claim 10 whereinthe step of splitting comprises producing maximally entangled photons.20. The method of claim 10 wherein the step of performing a plurality ofcoincidence measurements comprises performing a plurality of coincidencemeasurements between entangled photon pairs produced by a componentselected from the group consisting of: the first nonlinear crystal andthe second nonlinear crystal.
 21. The method of claim 10 wherein thesteps of splitting and directing comprise producing multiply entangledphotons.
 22. The method of claim 10 wherein the first signal comprisessynthetic aperture radar information.
 23. The method of claim 10 whereinthe step of performing a plurality of coincidence measurements comprisesperforming a plurality of coincidence measurements for different valuesof a parameter selected from the group consisting of: a polarizationstate, a transverse spatial position, a longitudinal spatial position, adelay, an a quantum state produced by a non-local operation.
 24. Themethod of claim 10 further comprising selecting the second signal,whereby the parameter associated with the first signal is at leastpartially determined by the second signal.
 25. A method for processingcomplex signals comprising: providing light; imposing a first signal ona first polarized component of the light to produce first encoded light;imposing a second signal on a second polarized component of the light toproduce second encoded light; transmitting the first encoded light andthe second encoded light through adjacent nonlinear crystals separatedby a distance; and determining a property of one of the first signal andthe second signal using results of at least four coincidencemeasurements of entangled photons.
 26. A method of encoding classicalinformation as a quantum state, the method comprising: producing light;separating the light into a first polarized component and a secondpolarized component; modulating the first polarized component with afirst classical signal to produce first modulated light; modulating thesecond polarized component with a second classical signal to producesecond modulated light; and directing the first modulated light and thesecond modulated light through a first downconverter and a seconddownconverter.